A323728 - OEIS

%I #23 Mar 20 2024 23:09:42

%S 2,5,10,8,26,13,50,20,18,29,122,25,170,53,34,32,290,45,362,41,58,125,

%T 530,52,50,173,90,65,842,61,962,80,130,293,74,72,1370,365,178,89,1682,

%U 85,1850,137,106,533,2210,100,98,125,298,185,2810,117,146,113,370

%N a(n) is the smallest number k such that both k-2*n and k+2*n are squares.

%C When n is a prime number, a(n) is greater than all the previous terms.

%C If n = 4*x*y, then a(n) is the smallest integer solution of the form 4*(x^2 + y^2), with rational values x and y.

%H Robert Israel, <a href="/A323728/b323728.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Difference_of_two_squares">Difference of two squares</a>

%F a(n^2) = 2 * n^2.

%F a(p) = p^2 + 1, for p prime.

%F a(n) = A063655(n)^2 - 2*n.

%F a(n) = A056737(n)^2 + 2*n.

%F a(n!) = A061057(n)^2 + 2*n!.

%F a(n) = A033676(n)^2 + A033677(n)^2. - _Robert Israel_, Feb 17 2019

%F a(n) = Min_{d|n} ((n/d)^2 + d^2). - _Ridouane Oudra_, Mar 17 2024

%e For n = 3, a(3) = 10, which is the smallest integer k such that k+2*n and k-2*n are both squares: 10+2*3 = 4^2 and 10-2*3 = 2^2.

%e For n=1..10, the following {a(n)-2*n, a(n)+2*n} pairs of squares are produced: {0, 4}, {1, 9}, {4, 16}, {0, 16}, {16, 36}, {1, 25}, {36, 64}, {4, 36}, {0, 36}, {9, 49}.

%p f:= proc(n) local d;

%p d:= max(select(t -> t^2 <= n, numtheory:-divisors(n)));

%p d^2 + (n/d)^2

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Feb 17 2019

%t Array[Block[{k = 1}, While[Nand @@ Map[IntegerQ, Sqrt[k + 2 {-#, #}]], k++]; k] &, 57] (* _Michael De Vlieger_, Feb 17 2019 *)

%o (PARI) a(n) = for(k=2*n, oo, if(issquare(k+2*n) && issquare(k-2*n), return(k)));

%o (PARI) a(n) = my(d=divisors(n)); vecmin(vector(#d, k, 4*((d[k]/2)^2 + (n/d[k]/2)^2)));

%Y Cf. A000290, A029744, A033676. A033677, A056737, A061057, A063655, A087711.

%K nonn

%O 1,1

%A _Daniel Suteu_, Jan 25 2019